Sequences and Series of Real Numbers and Functions

Overview

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Dive deep into the world of sequences and series of real numbers and functions in this comprehensive 4.5-hour tutorial. Explore fundamental concepts such as convergence, bounds, and monotonicity of sequences. Learn about the Bolzano-Weierstrass Theorem, Cauchy sequences, and limit supremum and infimum. Investigate sequences of functions, focusing on pointwise and uniform convergence. Delve into Taylor series, geometric series, and various convergence tests for series of real numbers. Examine the Cauchy-Schwarz inequality, Weierstrass M-test, and Mertens' Theorem. Gain insights into conditional and absolute convergence, Cauchy products, and Stirling's Formula. Conclude with practical examples involving limit supremum and infimum, and explore sequences that limit to zero while their series diverge.

Syllabus

Adding, Multiplying, and Dividing Convergent Sequences of Real Numbers.
Bounds on a Convergent Sequence of Real Numbers.
Sandwiching Sequences of Real Numbers.
Monotone Sequences of Real Numbers.
Subsequences of Real Numbers.
Bolzano Weierstrass Theorem.
Cauchy Sequence of Real Numbers.
Limit Supremum and Limit Infimum of a Sequence of Real Numbers.
Sequences of Functions - Pointwise and Uniform Convergence.
Uniform Convergence of Continuous Functions.
Uniform Convergence of Riemann Integrable Functions.
Taylor Series - Integral Form of the Remainer.
Geometric Series.
Prove that the Sequence {(1+1/n)^n} is Convergent using Sequence Theory.
Uniform Convergence of Bounded Functions.
Uniform Convergence of Differentiable Functions.
Series of Real Numbers.
Comparison Test for Convergence of a Series.
Cauchy–Schwarz Inequality.
Weierstrass M Test.
Ratio Test for Convergence of a Series.
Root Test for Convergence of a Series.
The Ratio Test Implies The Root Test.
Integral Test for Convergence of a Series.
Conditional and Absolute Convergence.
Cauchy Product.
Mertens' Theorem - Product of 2 Infinite Series.
Limit Supremum and Limit Infimum of Sets (part 1 of 2).
Limit Supremum and Limit Infimum of Sets (part 2 of 2).
Proof of Stirling's Formula for an approximation of n!.
Sequences that Limit to Zero, But the Series Diverges. #TeamSeas.
2 Examples with limsup and liminf.